NIMCET 2010 — Mathematics PYQ

1. Let the required value be $x$:

$\cos \theta -4\sin \theta =1\text{—(Eq 1)}$

$4\cos \theta +\sin \theta =x\text{—(Eq 2)}$

2. Square and add both equations:

$(\cos \theta -4\sin \theta {)}^2+(4\cos \theta +\sin \theta {)}^2=1^2+x^2$

3. Expand the terms:

$({\cos }^2\theta +16{\sin }^2\theta -8\sin \theta \cos \theta )+(16{\cos }^2\theta +{\sin }^2\theta +8\sin \theta \cos \theta )=1+x^2$

4. Simplify:

$17{\cos }^2\theta +17{\sin }^2\theta =1+x^2$

$17({\cos }^2\theta +{\sin }^2\theta )=1+x^2$

Since ${\cos }^2\theta +{\sin }^2\theta =1$:

$17(1)=1+x^2$

$17-1=x^2$

$x^2=16$

$x=\pm 4$

Correct Option:

(d) ±4